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Review Article
Fractal analysis of remotely sensed images: A review of methods andapplications
W. SUN*{, G. XU{, P. GONG§ and S. LIANG"
{Department of Geography and Planning, Grand Valley State University, Allendale,
MI 49401, USA. e-mail: sunwa@gvsu.edu
{Department of Geography and Planning, Grand Valley State University, Allendale,
MI 49401, USA. e-mail: xug@gvsu.edu
§Department of Environmental Science, Policy & Management, University of California,
Berkeley, 151 Hilgard Hall, CA94720, USA. e-mail: gong@nature.berkeley.edu
"Department of Geography, University of Maryland, College Park, MD 20742, USA.
e-mail: sliang@geog.umd.edu
(Received 12 June 2005; in final form 8 March 2006 )
Mandelbrot’s fractal geometry has sparked considerable interest in the remote
sensing community since the publication of his highly influential book in 1977.
Fractal models have been used in several image processing and pattern
recognition applications such as texture analysis and classification.
Applications of fractal geometry in remote sensing rely heavily on estimation
of the fractal dimension. The fractal dimension (D) is a central construct
developed in fractal geometry to describe the geometric complexity of natural
phenomena as well as other complex forms. This paper provides a survey of
several commonly used methods for estimating the fractal dimension and their
applications to remote sensing problems. Methodological issues related to the use
of these methods are summarized. Results from empirical studies applying fractal
techniques are collected and discussed. Factors affecting the estimation of fractal
dimension are outlined. Important issues for future research are also identified
and discussed.
1. Introduction
Fractal geometry was introduced and popularized by Mandelbrot (1977, 1982) to
describe highly complex forms that are characteristic of natural phenomena such as
coastlines and landscapes. The main attraction of fractal geometry stems from itsability to describe the irregular or fragmented shape of natural features as well as
other complex objects that traditional Euclidean geometry fails to analyse. In this
sense, fractal geometry provides a new language in which previously intractable
natural features can be described with more mathematical rigor (Barnsley 1989).
Clarke and Schweizer (1991:p.37) note that ‘Fractal geometry has been called one of
the four most significant scientific concepts of the 20th century, on a par with
quantum mechanics, the general theory of relativity, and the double-helix model of
the structure of DNA.’
Fractal geometry has sparked considerable interest in the remote sensingcommunity since the publication of Mandelbrot’s book, Fractals: Form, Chance
and Dimension, in 1977. The relevance and usefulness of fractal geometry to solving
remote sensing problems can be attributed to the fact that remotely sensed images
International Journal of Remote Sensing
Vol. 27, No. 22, 20 November 2006, 4963–4990
International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online # 2006 Taylor & Francis
http://www.tandf.co.uk/journalsDOI: 10.1080/01431160600676695
are not only spectrally and spatially complex, but they often exhibit certain
similarities at different spatial scales (Lam and De Cola 1993a). This requires us to
examine spatially complex patterns with relatively simple indicators such as various
measures of texture. It has been recognized that remotely sensed data can be
analysed using five types of signature: spectral, spatial, temporal, angular, and
polarization (Liang 2004). As more and more high spatial resolution imagery
becomes available, utilization of spatial signatures plays an increasingly important
role in extracting land surface properties from remotely sensed data. How to extract
the complex and erratic textures in the image and use spatial information to improve
image understanding and classification has been a major research issue in remote
sensing for decades (Haralick et al. 1973, Weszka et al. 1976, Pratt et al. 1978, Gong
and Howarth 1990, Wang and He 1990, Gong et al. 1992, Tso and Mather 2001). In
this context, fractal geometry appears especially appealing because it offers
something important, that is, tools for characterizing complex objects and land
surface patterns in remotely sensed images.
Fractal models have been used in a variety of image processing and pattern
recognition applications. For example, several researchers have applied fractal
techniques to describe image textures and segment various types of images (Pentland
1984, Keller et al. 1989, De Jong and Burrough 1995, Myint 2003). Fractal
characterization of the ‘roughness’ of remotely sensed images has been considered
useful as part of the metadata of images or as a tool for data mining or change
detection (Lam 1990, Jaggi et al. 1993, Emerson et al. 2004). Fractal models have
also been used to study the scaling behaviour of geographic features and the
knowledge generated by this type of research may be valuable for determining the
optimum resolution of pixels and polygons used in remote sensing and GIS
applications (Goodchild 1980, Lovejoy 1982, Mark and Aronson 1984, Emerson
et al. 1999).
Applications of fractal techniques to image analysis rely heavily on the estimation
of fractal dimensions. The fractal dimension, often denoted D, is a key parameter
developed in fractal geometry to measure the irregularity of complex objects. A
variety of methods have been proposed to compute the D of features such as
topographic surfaces and image intensity surfaces. However, most computational
methods have their theoretical and/or practical limitations. Several studies (Roy
et al. 1987, Tate 1998, Lam et al. 2002, Sun 2006, Sun et al. 2006) have reported that
different methods often yield significantly different D values for the same feature. In
addition to method-induced errors, a number of other factors such as the choice of
input parameter values and image data used may also influence computed D values.
As such, there is considerable uncertainty regarding the nature and extent of
variations in computed D.
Despite these developments, there has been no review paper summarizing and
evaluating different methods and applications that are widely scattered in the
literature. The purpose of this paper is to provide a survey of commonly used
methods for estimating fractal dimension and their applications to the analysis of
remotely sensed images. The focus of our discussion is on the methodological issues
related to the practical measurement of D in the remote sensing context. Contrasting
or conflicting results from empirical studies are collected and discussed. Major
factors influencing the computed D are outlined. Important issues for future
research are also identified and discussed. The next section provides a brief
introduction to the basic concepts of fractal geometry, followed by a description of
4964 W. Sun et al.
six computational methods. Examples illustrating applications of fractal techniques
in remote sensing are then presented. Major issues encountered in fractal analysis of
remotely sensed images are finally discussed. We conclude the paper with some
general remarks.
2. Fractals, self-similarity, and the fractal dimension
‘Clouds are not spheres, mountains are not cones, coastlines are not circles, and
bark is not smooth, nor does lightning travel in a straight line…’ (Mandelbrot
1982:p.1). Nature is complex. Many important features and patterns of nature are so
irregular that classical Euclidean geometry is hardly of any help in describing their
form. It was this inability of classical geometry to describe the real world that led
Mandelbrot (1977) to invent the concept of ‘fractal’ to fill the void caused by the
absence of suitable geometric representation for a family of shapes that are
continuous but not differentiable.
According to Mandelbrot (1977:p.4), the term fractal comes from the Latin
adjective fractus, which is also the root for fraction and fragment and means
‘irregular or fragmented’. Formally, a fractal is defined as a set for which the
Hausdorff-Besicovitch (or fractal) dimension strictly exceeds the topological
dimension (Mandelbrot 1977). A fundamental characteristic of fractal objects is
that their measured metric properties, such as length or area, are a function of the
scale of measurement. A classical example to illustrate this property is the ‘length’ of
a coastline (Richardson 1961, Mandelbrot 1967). When measured at a given spatial
scale d, the total length of a crooked coastline L(d) is estimated as a set of N straight-
line segments of length d. Because small details of the coastline (e.g. peninsulas) not
recognized at lower spatial resolutions become apparent at higher spatial
resolutions, the measured length L(d) increases as the scale of measurement d
increases. Thus, in fractal geometry, the Euclidean concept of ‘length’ becomes a
process rather than an event, and this process is found to be controlled by a constant
parameter (Richardson 1961). Mandelbrot (1967, 1977) generalized and expanded
on Richardson’s (1961) empirical findings and showed that the relationship between
length and measuring scale can be described by the power law:
L dð Þ~Kd 1{Dð Þ, ð1Þ
where the exponent D is called the fractal dimension, and K is a constant.
The scaling exponent D in equation (1), i.e. the fractal dimension, is a central
construct of fractal geometry. It is called fractal dimension because it is a fractional
(or non-integer) number (Mandelbrot 1977). The idea of using D to describe
irregular shapes is a powerful one because it captures what is lost in traditional
geometrical representation of form. In Euclidean geometry, dimensions are integers
or whole numbers (e.g. 1 for lines, 2 for areas, and 3 for volumes), and topological
dimensions remain constant no matter how irregular a line or an area may be. Thus,
a straight line and a crooked coastline have the same topological dimension 1, and a
smooth surface and a rugged topographic surface have the same topological
dimension 2. In other words, topology cannot discriminate between crooked lines
and straight lines (Mandelbrot 1982). As such, part of the information about the
form of irregular objects is necessarily lost in topological representations.
In fractal geometry, on the other hand, dimension is treated as a continuum. A
curve’s dimension, for example, can take on any non-integer value between 1 and 2,
Fractal analysis of remotely sensed images 4965
depending on the degree of irregularity of its form. The more contorted a line is, the
higher its dimension. Similarly, a surface’s dimension may be a non-integer value
between 2 and 3. The use of a fractional power in the description of complex shapes
compensates, in effect, for the length or area lost because of details smaller than the
measurement scale (d). With D it becomes possible to obtain consistent estimates of
an object’s metric properties at different measurement scales (Pentland 1984).
Fractal dimension can be thought of as a measure of an object’s ability to ‘fill’ the
space in which it resides. A smooth line of D51 will approach D52 when it becomes
so complex that it effectively takes up the whole plane. Similarly, as a surface’s D
approaches the upper value 3, it will appear increasingly rugged and display a rapid
succession of peaks and valleys. More generally, the more irregular an object
becomes, the more space it fills, and the higher its D value. In this way, the value of
D is intimately linked to our notion of ‘complexity’ or ‘roughness’ (Pentland 1984).
Self-similarity is another key property of fractals. Formally, self-similarity is
defined as a property where a subset, when magnified to the size of the whole, is
indistinguishable from the whole (Mandelbrot 1977, Voss 1988). The property of
self-similarity implies that the form of an object is invariant with respect to scale. In
other words, a strictly self-similar object can be thought of as being constructed of
an infinite number of copies of itself. In the geosciences, the property of self-
similarity may be better termed scale-independence (Clarke 1986). The forms of
natural phenomena are often erratic as ‘chances’ or random factors often play an
important role in their generating processes (Mandelbrot 1977). As such, unlike
mathematical fractals, natural objects generally do not display exact self-similarity.
Instead, they may exhibit a certain degree of statistical self-similarity over a limited
range of scales. Statistical self-similarity refers to scale-related repetitions of overall
complexity, but not of the exact pattern (Voss 1988).
Self-similar objects are isotropic (or rotation invariant) upon rescaling. If
rescaling of an object is anisotropic, then the object is said to be self-affine.
Formally, with self-affine fractals the variation in one direction scales differently
than the variation in another direction (Mandelbrot 1985). Thus, the trail of
particulate Brownian motion in two-dimensional space is self-similar, whereas a plot
of the x-coordinate of the particle as a function of time is self-affine (Brown 1995).
Similar to the concept of statistical self-similarity, an object is said to be statistically
self-affine if it displays self-affinity only in a statistical sense.
Fractals, self-similarity, and fractal dimension are the key concepts of fractal
geometry upon which most remote sensing applications seem to have drawn. The
relevance of these concepts to the analysis of remotely sensed images will be
discussed in greater detail in the following sections. The reader is referred to
Mandelbrot (1977, 1982) for a more complete discussion of fractal geometry. For an
introduction to fractal analysis of images, the reader is directed to Peitgen and
Saupe (1988). An excellent introduction to fractals in geography can be found in
Lam and De Cola (1993a). The review of fractals in physical geography presented by
Gao and Xia (1996) is also informative.
It should be noted that, although this paper focuses on the current state of fractal
analysis techniques in Earth imaging, fractal geometry has found application in a
wide range of scientific fields (Dyson 1978). For example, fractal models have been
used extensively in pattern recognition (e.g., Peleg et al. 1984, Dennis and Dessipris
1989, Chaudhuri et al. 1993, Blacher et al. 1993). Fractal geometry has contributed
much to computer science (e.g. Peitgen and Richter 1986, Devaney and Keen 1989).
4966 W. Sun et al.
In computer graphics, fractal techniques have been used, for example, to simulate
realistic landscapes such as rugged terrains, which can be used in motion pictures
and flight simulators (Fournier et al. 1982). Fractal geometry has also been applied
to such diverse fields as meteorology (Lovejoy and Schertzer 1985, 1990), ecology
(Loehle 1983, Wiens 1989), material science (Lu and Hellawell 1995), urban
landscapes (Batty & Longley 1986), economics and finance (Calvet and Fisher 2002,
Mandelbrot and Hudson 2004), soil sciences (Burrough 1981, Armstrong 1986,
Green and Erskine 2004), and medical imaging (Chen et al. 1989, Wu et al. 1992,
Lee et al. 2003). Readers interested in the works done outside the Earth imaging
realm should consult major journals in fields of interest.
3. Methods to compute the fractal dimension of image intensity surfaces
The fractal dimension of strictly self-similar objects can be derived mathematically
and is given by (Mandelbrot 1977):
D~log Nrð Þlog 1=rð Þ , ð2Þ
where Nr represents an object of Nr parts scaled down by a ratio of r. The D derived
from equation (2) is called the shape’s similarity dimension (Mandelbrot 1977).
For non-mathematical objects, however, the fractal dimension cannot be derived
analytically. Instead, it must be empirically estimated. A large number of methods
have been proposed to compute the fractal (monofractal) dimension of natural
objects. These methods differ in the ways they approximate the quantity Nr in
equation (2), but they are similar in spirit in that they use some version of the
statistical relationship between the measured quantities of an object and step sizes to
derive the estimates of D. The ‘quantity’ of an object is expressed in terms of, for
example, length, area, or number of boxes (cells) needed to cover the object. ‘Step
size’ refers to the scale or resolution of measuring units used. The procedure
common to most of the methods discussed in this paper consists of three steps:
N First, measure the quantities of the object under consideration using various
step sizes.
N Second, plot log (measured quantities) versus log (step sizes) and fit a least-
squares regression line through the data points. The log–log plot is often
referred to as the Richardson plot.
N Third, use the slope of the regression line to derive the D of the object.
A remotely sensed image can be viewed as a hilly terrain surface whose ‘elevation’
is proportional to the grey level or digital number (DN) value. As such, all methods
developed to compute the D of surface features can, in principle, be readily applied
to remotely sensed images. Broadly, there are two basic approaches for computing
the D of surfaces. The first is to directly estimate D from the surfaces being analysed.
The second approach involves the so-called dimensionality-reduction technique
(Klinkenberg 1994). In this approach, the D of a surface is estimated by first
calculating the D of contours or profiles extracted from the surface and, then, simply
adding 1 to account for the different Euclidean dimension. In this section, we
describe six methods for computing the D of surface features such as remotely
sensed images. For a review of the methods used to calculate the D of linear features,
the reader is directed to Klinkenberg (1994).
Fractal analysis of remotely sensed images 4967
3.1 The triangular prism method
The triangular prism method was developed by Clarke (1986) primarily to calculate
the D of topographic surfaces, but it has been applied extensively to remotely sensed
images. The method makes use of a raster representation of the elevations of the
Earth’s surface such as in a digital elevation model (DEM) (see figure 1 (a and b)).
Based on this data structure, the method takes elevation values (or the equivalent of
DN value in an image) at the corners of squares (i.e. analysis windows) (a, b, c and d
in figure 1a), interpolates a centre value (e in figure 1(a)), divides the square into four
triangles (abe, bce, cde and dae in figure 1(b)), and then computes the top surface
areas of the prisms which result from raising the triangles to their given elevations
(A, B, C, and D in figure 1(a)). By repeating this calculation for geometrically
increasing square sizes (d), the relationship between the total upper surface area of
the prisms (i.e. the sum of areas A, B, C, and D in figure 1(a)) and the spacing of the
squares (i.e. step size d) can be established, and used to estimate D (table 1). The only
input parameter required by this method is the number of step sizes. Generally, the
method is less computationally intensive than other methods such as the variogram
or Fourier power spectrum methods (Clarke 1986).
Several modifications have been proposed since Clarke (1986) presented his
method. In Clarke’s original algorithm, ‘spacing of the square’ was interpreted as
‘area of the square.’ In other words, Clarke used step size squared (d2) in the
Figure 1. (a) 3D view of the triangular prism method (after Clarke, 1986); (b) Top view ofthe corner pixels (a, b, c, and d ) and the centre point (e) used in Clarke’s (1986) method (anexample with step size 5 4).
4968 W. Sun et al.
Table 1. Methods for computing the fractal dimension (D) of surface features{.
Method Relation used Basic formula Estimate of D
Triangular prism Total area of the tops ofprisms vs. side length of analysiswindows
S(d)jd22D Plot log S(d) versus log (d),S(d)5area slope is (22D)d5side length of analysis windows D522slope
Differential box counting Number of boxes needed to coveran image vs. box size
Nrj(1/r)2D Plot log Nr versus log (1/r),Nr5number of boxes Slope is 2Dr5s/M,s5side length of boxes, D52slopeM5side length of the image
Variogram Mean squared elevation (or DN)difference vs. distance
E[(Zp2Zq)2]j(dpq)2H Plot log {E[…]} versus log dpq
Zp, Zq5elevations or DNs at points pand q
slope is 2H
dpq5distance between p and q D532H532slope/2Isarithm Length of contour line vs.
step sizeL(d)jd12Dcontour For each contour line, plot log L(d)
versus log (d),L(d)5length of contour line (i.e., numberof boundary cells)
slope is (12Dcontour)
d5step size D5average of all Dcontour + 1Robust fractal estimator Length of profile vs. step size L(d)jd12Dprofile For each cell, take the average of Dprofile
in both EW and NS directionsL(d)5length of profile D obtained by combining fractal dimen-
sions of each cell using a weightedaverage and adding 1
d5step sizePower spectrum Fourier power spectral
density vs. the frequencyP(f)jf2(522Dprofile) Plot log P(f) versus log (f),P(f)5power, slope is – (522Dprofile)f5frequency D5Dprofile + 1
{Dcontour5fractal dimension of a contour line; Dprofile5fractal dimension of a profile. For references see §3 of the text.
Fra
ctal
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aly
siso
frem
otely
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49
69
regression. Lam et al. (2002) have shown that step size (d) instead of step size
squared (d2) should be used to derive the correct D. Sun (2006) has recently
proposed three new procedures to implement the triangular prism method. Sun’s
methods differ from Clarke’s method in that, in constructing the prisms, they take
into account the actual DN values of all the pixels at the edges of an analysis
window. Sun (2006) reported that her methods perform better than Clarke’s (1986)
original method when applied to images with complex textures.
Clarke (1986) reported that the triangular prism method provided good results for
the test data used in his work. Clarke and Schweizer (1991) found that subsequent
tests of the method on terrain data have yielded rather low D values. In a systematic
study comparing the performance of several methods, Lam et al. (2002) showed that
the triangular prism method was the best estimator for rougher surfaces with
generated D52.9 and 2.7, although it was less accurate where generated D52.5 and
2.3. They also found that the triangular prism method was sensitive to contrast
stretching and recommended that, to ensure comparability and accuracy of
measurement, the range of DN values of an image be normalized before using the
method. The triangular prism method was also found to be sensitive to ‘noise’ or
extreme grey level values (Qiu et al. 1999). To obtain reliable results, it is important
first to assess whether the image to be measured has any noisy pixels before applying
the method.
3.2 The differential box-counting (DBC) method
The differential box-counting (DBC) method was proposed by Sarkar and
Chaudhuri (1992) to compute the D of digital images. This method can be thought
of as a variant of the well-known box-counting approach (Goodchild 1980, Voss
1988). In the differential box-counting method, Nr in equation (2) is counted in the
following manner. If an image of size M6M pixels is scaled down to a size s6s
where M/2>s.1 and s is an integer, then we have a ratio of r5s/M. Consider the
image as a 3D space with (x, y) denoting the image plane and (z) denoting the grey
level. The (x, y) space is partitioned into grids of size s6s. On each grid there is a
column of boxes of size s6s6s9. If G denotes the grey level range of the image (e.g.
256), s9 is calculated by [G/s9]5[M/s]. Let the minimum and maximum grey level
of the image in (i, j)th grid fall in box number zmin and zmax, respectively. Then
nr(i, j)5zmax2zmin + 1 is the contribution of Nr in (i, j)th grid. Taking contributions
from all grids, we have:
Nr~X
i,j
nr i, jð Þ ð3Þ
For different values of r, that is, different values of s or step sizes, the quantity Nr is
counted. D then is computed from the least-squares linear fit of log (Nr) versus log
(1/r) (table 1).
Sarkar and Chaudhuri (1992) have shown that the DBC method is both accurate
and computationally efficient. Despite this, the method does not seem to have been
widely applied to remote sensing problems (Tso and Mather 2001). One possible
explanation for the lack of interest in this method among remote sensing researchers
is that the method was developed outside the geosciences domain. It remains to see
how effective the DBC method is when applied to remotely sensed images.
4970 W. Sun et al.
3.3 The variogram method
The variogram method is a widely used technique for computing D of surfaces. In
this method, the mean of the squared elevation (or DN) difference (i.e. variance) is
calculated for different distances, and D is estimated from the slope (b) of the
regression between the logarithms of variance and distance (see figure 2) so that
D532b/2 (Mark and Aronson 1984; see also table 1). Variations of the variogram
method exist. Roy et al. (1987), for example, calculated D using four different
implementations of the variogram method.
Because of its ease of use, the variogram method has been used in numerous
studies (Goodchild 1980, Burrough 1981, Mark and Aronson 1984, Roy et al. 1987,
Carr and Benzer 1991, Klinkenberg and Goodchild 1992). Lam and De Cola
(1993b) noted that three other properties of the variogram method may have also
contributed to its popularity. First, the method can be applied to both regular and
irregular data. Second, for irregular polygonal data, the variogram function can be
determined by centroids representing the polygons. Third, when compared with the
isarithm method (discussed in §3.4), the variogram method is generally more
reliable.
The variogram method is based on the assumption that the surface being analysed
is a fractional Brownian surface. Studies of natural terrain surfaces (Mark and
Aronson 1984, Roy et al. 1987) have shown, however, that variograms often do not
behave linearly at all scales, suggesting that natural phenomena are not truly fractal
(discussed in §5.1). Another issue that requires attention in variogram analysis is the
sampling strategy used to determine the point pairs. Klinkenberg (1994) suggested
that, for statistical reasons, the sample of point pairs uniformly span the distance
range of a given dataset and the distance classes be constructed so that they are
evenly spaced in the log–log space used in the regression. Furthermore, while the
shortest point-pair distance will depend on the resolution of the data used, attention
should be given to the choice of the maximum point-pair distance. The common
practice is to use one-half of the absolute maximum distance between points of a
given dataset as the maximum point-pair distance. Roy et al. (1987) have suggested
using even shorter maximums, such as one quarter the maximum distance.
Klinkenberg (1994) argued, however, that the rule of using one-half or one quarter
of the maximum distance as the maximum point-pair distance appears more
restrictive than necessary.
Figure 2. The log (variance) versus log (distance) plot used in the variogram method.
Fractal analysis of remotely sensed images 4971
Different studies reached different conclusions regarding the performance of the
variogram method. Klinkenberg and Goodchild (1992), for example, found that the
variogram method was able to produce consistent estimates of D when applied to
topographic data. Clarke and Schweizer (1991), on the other hand, reported that for
the same dataset, the variogram method appears to yield consistently higher D
values than those obtained from the triangular prism method and the robust fractal
estimator (discussed in §3.5). The work of Lam et al. (2002) suggests that the
variogram method was a comparatively poor estimator for all the simulated surfaces
used in their study. One major limitation of the variogram method is that remotely
sensed images are data rich and exhaustive. In order to make the variogram method
computationally tractable, studies applying the variogram method to image analysis
use random sampling of a subset of pixels. This makes the computed D a random
variable that changes from one analysis to the next.
3.4 The isarithm method
The isarithm method (Shelberg et al. 1983) is based on the premise that the
complexity of isarithm or contour lines may be used to approximate the complexity
of a surface. Briefly, the method works in the following way. Starting with a matrix
of z-elevations (or DN values), an isarithm interval is selected and isarithm lines are
constructed on the surface. For each isarithm line, its lengths are calculated in terms
of the number of boundary cells over a number of step sizes, log (number of
boundary cells) is regressed against log (step sizes), and the slope of the regression
line is used to derive the D of the isarithm line. This process is repeated for every
isarithm line. The surface’s D is obtained by averaging the D values of all the
isarithm lines that have R2>0.9 and adding 1 (table 1).
To implement the isarithm method, a data matrix of a given number of rows and
columns must be specified, with the following parameter input by the user: (1) the
number of step sizes, (2) the isarithm interval, and (3) the direction in which the
computation is implemented (row, column or both). It is possible that for a given
step size, there are no boundary cells. In this case, the isarithm line is excluded from
the analysis to avoid regression using fewer points than the given number of steps
(Shelberg et al. 1983). Lam (1990) pointed out that this feature appears especially
useful for the analysis of remotely sensed images as it ensures that random noise in
the image will not be taken into account in the estimation process.
Lam and De Cola (1993b) have discussed several factors that may influence the
computed D values using the isarithm method. They noted that real data are
generally anisotropic and, therefore, the computed D will vary depending on
whether it is measured along rows, columns, or in a non-cardinal direction. The
maximum step size used may also affect the reliability of estimation results.
Furthermore, the practice of using the average of the D values of those isarithm lines
that have R2>0.9 to represent the surface D is somewhat arbitrary.
Shelberg et al. (1983) pointed out that an advantage of the isarithm method is that, by
using a number of isarithm lines, the method can be used to estimate the D for non-self-
similar surfaces. Furthermore, the method was found to be robust to random noise in
the image (Qiu et al. 1999). Clarke (1986), however, seemed quite critical of the isarithm
method and commented that, ‘This method is rather crude, however, and really is an
empirical estimate of an empirical estimate’ (p.714). Despite this, the isarithm method
appears to be one of the most often used methods for computing the D of remotely
sensed images (Lam 1990, Lam and De Cola 1993b, Qiu et al. 1999, Emerson et al. 1999,
4972 W. Sun et al.
Lam et al. 2002). Applications of the isarithm method have shown that the method
returned good results for images with medium-ranged complexity, but it overestimated
D when applied to rougher surfaces while it underestimated D for smooth images
(Emerson et al. 1999, Lam et al. 2002).
3.5 The robust fractal estimator
The robust fractal estimator was proposed by Clarke and Schweizer (1991) in an
attempt to provide stability in the computation of D. Using the walking-dividers
method, the robust fractal estimator computes for each cell the D of each profile in
both the east–west and north–south directions and places the average of the two in a
new array. The D of the entire surface is obtained by combining the D values of each
cell using a weighted average and adding one (Clarke and Schweizer 1991).
Clarke and Schweizer (1991) noted that the robust fractal estimator is primarily
designed to calculate D for natural surfaces using data from USGS DEMs, but it
should work equally well on any gridded surface data. Applications of this estimator
to image analysis seem limited, however. As such, little is known about its
performance. In their paper, Clarke and Schweizer reported that for the same
datasets used in their study, the robust fractal estimator appeared to consistently
yield a lower D value than those obtained from the triangular prism and variogram
methods. In a discussion of how to select the largest step size when using the divider
method on self-affine curves, Klinkenberg (1994) pointed out that the low estimated
D values (close to one) reported in Clarke and Schweizer (1991) may be due to the
fact that they used steps sizes that spanned the crossover length. Note that the
crossover length refers to the range of scale within which the computed D is
representative of the local fractal dimension of a self-affine feature. Discussion of
the crossover length concept can be found in Mandelbrot (1985).
3.6 The Fourier power spectrum method
Another technique for computing the D of surface features is the use of Fourier
analysis. The Fourier method uses the power spectrum derived from the surface
(Pentland 1984, Burrough 1981). It can be shown that the Fourier power spectrum
P(f) of a fractional Brownian function (f) is proportional to f(22h21), where
h522Dprofile (Pentland 1984). The fractal dimension of the profile (Dprofile) is
obtained from the slope of the regression line of the log-log plot of P(f) versus f. The
D of the surface is computed as D5Dprofile + 1 (table 1). Detailed descriptions of the
steps required to perform a spectral analysis for fractal applications can be found in
Peitgen and Saupe (1988) and Turcotte (1992).
Spectral methods should only be applied to self-affine curves (i.e. profiles) since
they will always return a D51 for self-similar curves (Peitgen and Saupe 1988).
Several researchers (Fox and Hayes 1985, Clarke 1986, Carr and Benzer 1991) have
pointed out that, although a rigorous method, the power spectrum method involves
sophisticated data preprocessing and is computationally complex, a fact that
appears to have limited its applications. The inherent complexity of spectral
methods requires that they be carefully implemented (Klinkenberg 1994).
4. Applications of fractal techniques to remote sensing image analysis
There is a large amount of literature on the applications of fractal techniques to the
analysis of remotely sensed images. A main thrust in this application literature is the
Fractal analysis of remotely sensed images 4973
use of D to measure the roughness or textural complexity of land surface features. In
this paper, we focus on four major application areas: that is, the use of computed D
(1) to characterize the overall spatial complexity of an image, (2) to supply image
classification with textual information, (3) to describe the geometric complexity of
the shape of feature classes in a classified image, and (4) to examine the scaling
behaviour of environmental phenomena. We provide below a review of
representative works published in each of these four application areas.
4.1 Using D to characterize the overall spatial complexity of remotely sensed images
Perhaps the most obvious utility of fractal models in image analysis is the use of D
to characterize the overall textural complexity of remotely sensed imagery (Lam
1990, Qiu et al. 1999, Read and Lam 2002, Weng 2003). In these applications, only a
single D for the entire image is computed. Such a global D can be calculated for
remotely sensed data of different land cover types, sensors, and bands. Lam (1990),
for example, used the isarithm method to measure the spatial complexity of three
Landsat TM images representing three different land cover types in coastal
Louisiana. She found that the estimated D values of these TM surfaces were
generally higher than those of most real-world terrain surfaces. Among the three
land cover types, the highest D occurred in an urban area, followed by a complex
coastal area and a rural area. She also compared the D values of the three land cover
types across seven spectral bands and found that the D values of the same land cover
type turned out to be quite different in different bands. The urban landscape has its
highest D values occurring in bands 2 and 3, whereas the coastal and rural areas
both exhibit high D values in band 1. Lam (1990) noted that, although the three land
cover types examined in her study appeared to have different D values, the
difference in the average D values among the three land cover types was small
compared to the differences in overall D values among bands.
In an analysis of two AVIRIS (Airborne Visible Infra-Red Imaging Spectrometer)
images of the Los Angeles area, Qiu et al. (1999) found that the computed D values
for urban landscapes were higher than those for rural landscapes. A novel part of
this study was the systematic comparison of the computed D values of the two study
areas across the full spectral range of the hyperspectral images (224 bands). They
confirmed Lam’s (1990) finding that the textural complexity of the same land cover
type, expressed as D values, varied significantly across bands. They found that
higher contrast in D values between the urban and rural landscapes occurred in the
visible bands. They also reported unusually high D values (D.2.9) detected in the
spectral bands where signal-to-noise ratios were low. An important finding of Lam
(1990) and Qiu et al. (1999) studies is that the texture of a land cover type may be
better characterized by certain band(s) than by others. As such, identifying the
bands in which the contrast in computed D between different land cover types is
most distinct may be a necessary step in dealing with multispectral images.
Fractal techniques have also been used to describe spatial variations of
environmental phenomena along certain transects extracted from remotely sensed
images. In a study of the urban heat island effect in a Chinese city, Weng (2003)
applied fractal techniques to analyse the spatial variability of surface radiant
temperature along three profiles constructed from Landsat TM images. His results
suggest that variations in estimated D values along a profile can be linked to
underlying land cover types. He also compared D values across several years
and between different seasons of the year. His results show that information about
4974 W. Sun et al.
inter-temporal changes in D values was useful in understanding the increased
textural complexity of the thermal surfaces as well as the seasonal dynamics of
urban heat island effect.
Fractal characterization of the overall complexity of remotely sensed images has
been considered useful as part of metadata or as a tool for data mining and change
detection (Jaggi et al. 1993, Lam et al. 2002, Emerson et al. 2004). An advantage of
the fractal technique in these applications is that global D values can be computed
without the need to first classify the image. Given the rapidly increasing types and
volumes of remotely sensed data available today, quantitative assessment of the
spatial characteristics of various images may become a useful exercise in selecting
the right data for a particular application. Another potential utility of fractal
characterization of remotely sensed data is the use of D as an initial screening tool
for examining information content contained within different spectral bands.
Variations in D values across bands may be used as a guideline for identifying noisy
bands or for the selection of bands for image display, classification, and analysis
(Lam 1990, Qiu et al. 1999). Such information may be particularly valuable for
applications where texture is important.
4.2 Use of D as a texture measure to segment and classify images
The use of only spectral signatures to distinguish land cover types has proved
inadequate. Numerous studies have shown that classification results may improve if
additional information about spatial variations in pixel values is incorporated in the
classification procedure. Many techniques, such as co-occurrence matrices (Haralick
et al. 1973), local variance (Woodcock and Strahler 1987), wavelets (Mallat 1989)
and spatial autocorrelation statistics (Cliff and Ord 1973), have been proposed to
extract textural information from remotely sensed images.
Fractal techniques appear well suited to the analysis of textural features in
remotely sensed images, as the environmental features captured in the image are
often complex and fragmented (Burrough 1981, Lorimer et al. 1994). It has been
suggested that local variations in computed D can be used as texture measures to
segment images (Pentland 1984, Keller et al. 1987). The idea is that different land
cover types may have characteristic textures or roughness that could be described by
different D values. Ideally, if there were a one-to-one relation between the texture of
a land cover type and a unique D value, then the D could be viewed as the ‘fractal
signature’ of that land cover type and used to extract it from the image.
Pentland (1984) pioneered the use of fractal geometry in image texture analysis
and segmentation. In his 1984 paper, Pentland considered the image intensity
surface as a fractal Brownian function (fBf) and estimated D from Fourier power
spectrum of fBf. He segmented several types of images using computed D values and
achieved good results. He also found that the computed D was always stable over at
least 4 : 1 variations in scale, and most segmentations were stable over a range of
8 : 1. He concluded that fractal-based image segmentation appeared to be a powerful
technique (Pentland 1984).
Perhaps the best example illustrating the utility of fractal techniques in image
classification was presented by De Jong and Burrough (1995). In their study, De
Jong and Burrough proposed a so-called ‘local D algorithm,’ a method that can be
thought of as a local implementation of the triangular prism concept (Clarke 1986).
In the ‘local D algorithm,’ a kernel of 9 by 9 pixels is moved over the image and, at
each position of the kernel, a D is computed within the kernel, resulting in a new
Fractal analysis of remotely sensed images 4975
image file containing the estimated local D values. This new layer of D values was
then used as texture measures in the classification procedure. De Jong and Burrough
applied their method to the classification of six Mediterranean vegetation types in
two remotely sensed images. The results from the analysis of these two images seem
somewhat mixed. While the ‘local D algorithm’ appeared effective in separating five
of the six land cover types in a Landsat TM image, the method could not sharply
distinguish between any of the six land cover types when applied to an airborne
GER (Geophysical Environmental Research Imaging Spectrometer) image. This
poor result was explained by the poor quality of the GER image. They concluded
that, although local D values for TM imagery seemed to reflect the different land
cover types examined in their study, D values by themselves were insufficient for the
classification of TM images.
Other studies attempting to gauge the usefulness of fractal techniques for image
classification purposes include Jones et al. (1989), Keller et al. (1989), LaGro (1991),
De Jong (1993), Myint (2003) and Sun (2006), among others. In a comprehensive
study comparing the discriminatory power of several texture analysis methods, Myint
(2003) found that the spatial autocorrelation approach (Moran’s I and Geary’s C) was
superior to fractal approaches (isarithm, triangular prism, and variogram) and, in
some cases, simple standard deviation and mean value of the samples gave better
classification accuracies than all or some of the fractal techniques. His results also
show that the computed D values for the same image vary with the computational
method and spectral band used. He concluded that fractal-based textural discrimina-
tion methods are applicable but these methods alone may be ineffective in identifying
different land cover types in remotely sensed images.
It should be noted that, when used to analyse local tonal variations in the image
(i.e. local D values), fractal techniques provide meaningful results only for image
portions larger than the smallest step size used. In other words, texture variations at
scales smaller than the smallest step size will be overlooked in fractal analysis. This is
often referred to as the blurring effect. How to choose an ‘appropriate’ window size
and how to deal with the boundary effect, as well as the blurring effect, are two
important issues that deserve attention in computing local D values. These issues
will be discussed in greater detail in §5.5.
4.3 Fractal characterization of classified image features
Fractal analysis has been shown to be of descriptive value in the analysis of spatial
complexity of classified image features. Lovejoy (1982), for example, analysed the
perimeter–area relationship of rain and cloud areas identified from satellite and radar
images. He found that the degree of contortion of the perimeter of cloud regions could
be described by D51.35 over a range of cloud sizes from less than 1 km2 to over
106 km2. A more comprehensive example of fractal description of classified image
features was presented by De Cola (1989). In his study, De Cola used the perimeter–
area relationship to describe the shape of regions of eight land cover classes extracted
from a Landsat TM image of north-west Vermont. He showed that it was possible to
associate land cover types with D values. For example, the forests examined in his
study were characterized by high D and large regions, while agricultural activities had
large regions with D inversely related to the intensity of cultivation, and urban land-
use yielded small regions with relatively high D.
Fractal description of classified image features can provide useful descriptive
statistics for characterization of the aggregate feature classes. Knowledge about the
4976 W. Sun et al.
characteristic D values of different feature classes may be valuable in understanding
the processes that generate the phenomena under consideration (Lovejoy 1982).
Another utility of fractal description of classified image features is the use of such
data as input to GIS. De Cola (1989) demonstrated how the analysis of individual
land cover regions, such as the urban regions in his study, can provide a raster-based
GIS data structure and be used to investigate the location and description of
individual regions and to check the reliability of classification and labelling
processes.
4.4 Scaling characteristics of remotely sensed images
Most research published to date has suggested that real remotely sensed images are
not true fractals (discussed in §5.1). This finding is clearly in violation with the
assumption underlying the fractal model, i.e. the objects under consideration are
self-similar, at least statistically. Some researchers (e.g. Lam et al. 2002) have
argued, however, that lack of self-similarity in real remotely sensed data could be
used positively. Given that the estimated D is stable only over limited ranges of
scale, the behaviour of D can be used to study the effects of scale changes on image
properties. Several studies have examined the scaling properties of digital images
(Lovejoy 1982, Pentland 1984, Emerson et al. 1999, Lam et al. 2002, He et al. 2002).
A good example of the research in this direction was provided by Emerson et al.
(1999). In their study, Emerson et al. examined the effect of changing pixel size on
the computed D values of NDVI (Normalized Difference Vegetation Index) images
of two study areas. In the example of Huntsville Alabama, they found that the
estimated D values of NDVI images of agriculture, forest, and urban areas
responded differently to aggregation. The image of the agricultural area grew more
complex as the pixel size was increased from 10 to 80 m, while the forested area grew
slightly smoother and the complexity of the urban area remained approximately the
same. The analysis of the image data of the East Humboldt Range in Nevada
showed a more complex relation between pixel size and D and this relation changes
between seasons.
Emerson et al. (1999) pointed out that information about the scaling behaviour of
environmental phenomena may be valuable for the selection of an optimal
resolution for characterizing the phenomena under investigation. For example,
environmental phenomena that are more scale independent may require fewer data
or lower resolutions than those that are highly scale dependent. They also argued
that the response of D values to scale changes may be used as a guide to identify the
scale at which the processes that produced certain phenomena operate.
In addition to the four major application areas discussed above, there are other
applications that are not reviewed here. For example, fractal techniques have been
applied to image simulation (Pachepsky et al. 1997, Ricotta and Avena 1998,
Ricotta et al. 1998), image compression (Belloulata and Konrad 2002), image
denoising (Ghazel et al. 2003), image filtering (Germain et al. 2003), characterization
of the structures of hydrological basins (Maitre and Pinciroli 1999), and so forth.
5. Discussion and research need
Remotely sensed images with different textural characteristics are expected to have
different D values. However, differences in image texture are not the only factor
influencing the computed D (figure 3). This raises the question of what is actually
Fractal analysis of remotely sensed images 4977
captured in the computed D. In this section, we discuss the major sources of
limitations and uncertainty in the application of fractal techniques in remote
sensing. The focus of our discussion will be on several issues common to the
methods and applications discussed above. Following a brief discussion of each of
these issues, we offer our thoughts on the research potential for fractals in remote
sensing.
5.1 Are remotely sensed images fractal?
The self-similarity property underlying the fractal model predicts that for truly
fractal surfaces, the computed D should be constant at all scales, at all locations, and
in all directions. Numerous studies have shown that the estimated D values of most
natural phenomena are unstable with respect to scale, location, and/or orientation
(Mark and Aronson 1984, Roy et al. 1987, Klinkenberg and Goodchild 1992,
Burrough 1993). The consensus that has emerged from the research published to
date is that, as far as natural phenomena are concerned, self-similarity is exhibited
only in a statistical sense and such statistical self-similarity, when present, is
exhibited only in limited regions and over limited ranges of scale (Goodchild and
Mark 1987, Milne 1991).
Are remotely sensed images fractal? Studies directly addressing this issue appear
scarce. Nevertheless, a number of studies have found that the estimated D of real
remotely sensed images vary with the resolution of the image used and the region
and direction in which D was computed (Lam 1990, Emerson et al. 1999, De Jong
and Burrough 1995, Lam et al. 2002, Sun et al. 2006). De Jong and Burrough (1995)
Figure 3. Factors influencing the computed fractal dimension of remotely sensed images.
4978 W. Sun et al.
have further noted that the log–log plots constructed in their study were nonlinear
beyond a certain range, indicating breaks in the slope of the regression lines and
hence the D. These results suggest that most remotely sensed images are not strictly
self-similar; instead, they may be at most only statistically self-similar over a limited
range of pixel sizes.
The observation that most remotely sensed images may not be even statistically
self-similar brings up an important issue, i.e. does it make sense to use D to describe
image textures? At the theoretical level, lack of self-similarity does violate the
assumptions underlying most of the methods discussed in §3. For example, the
variogram method assumes that the surfaces being analysed have statistical
properties similar to those of fractal Brownian surfaces. If this were not the case,
then the method would not necessarily yield a correct D (Piech and Piech 1990).
However, some researchers have suggested that lack of self-similarity is not a
limitation to the fractal technique and it could be simply seen as a method for
extracting information from the Richardson plot (Orford and Whalley 1983,
Kennedy and Lin 1986, Normant and Tricot 1993).
The above discussion suggests that there is still considerable uncertainty regarding
to what extent remotely sensed images are (statistically) self-similar and whether
self-similarity is a prerequisite to applying fractal techniques. More research is
clearly needed in this area. For example, if one accepts that the fractal technique
could be used simply as a method to extract textural information, then it may be
argued that the structures under consideration do not have to be self-similar. But, do
existing computational methods, which operate on the assumption that the object
being measured is self-similar/affine, react differently to image textures than to
structural self-similarity/affinity in any significant ways? Very little has been written
about this issue in the remote sensing literature. Furthermore, if one accepts that
statistical self-similarity, when present at all, is exhibited only over limited ranges of
scale in real images, then one needs to consider if characterization of image textures
using a single (i.e. monofractal) dimension is adequate (discussed in §5.6).
Another issue worth exploring is the use of fractal models to detect edge points in
remotely sensed images. Given that breaks in D appears to be the norm rather than
exception in most real images, it should be possible to extract edge points by
identifying breakpoints in estimated D. Conceptually, such breakpoints could be
considered as the boundaries between homogenous regions with different textural
features (Pentland 1984). Research is needed to test such a fractal-based edge
detection method using a variety of images and compare it with other existing edge
detection algorithms to establish its performance.
5.2 Method-induced errors
The utility of D as a texture measure depends to a large extent on the reliability of
computational methods. A major difficulty in establishing the reliability of various
fractal computational methods is that the D values computed in different ways are
not necessarily related, unless there is a mathematical relationship among the
various D values. In fact, many studies have shown that different computational
methods often yield different results for the same data set (Lam 1990, Clarke and
Schweizer 1991, Klinkenberg and Goodchild 1992, De Jong and Burrough 1995,
Myint 2003, Sun 2006). The differences among different methods are often so
significant that comparisons between computed D values obtained using different
methods are meaningless. Klinkenberg and Goodchild (1992) have even argued that
Fractal analysis of remotely sensed images 4979
the variability in computed D is more a function of the methods used than it is a
reflection of any theoretical inadequacy of the fractal model.
Several factors may be responsible for the observed differences in estimated D
obtained using different methods (Klinkenberg 1994). First, some of the differences
in computed D may arise from the fact that fractal computational methods are not
all measuring the same fractal quantity. For example, while a stochastic relation is
used in the triangular prism method (Clarke 1986) to derive the estimates of D, the
relation used in those methods based on the walking-dividers approach, such as the
isarithm method (Shelberg et al. 1983) and the robust fractal estimator (Clarke and
Schweizer 1991) discussed above, is a geometric one. Second, part of the differences
in computed D may result from inappropriately applied methods. For example, the
robust fractal estimator (Clarke and Schweizer 1991) can only be applied to self-
affine data. Violation of this requirement could lead to erroneous results.
Furthermore, even when an appropriate method is chosen, whether the method is
executed properly may also affect the results. When working with self-affine data,
for example, computation would always return a D close to 1 if step sizes exceeded
the crossover length. Third, even when an appropriate method is used and properly
executed, the details of the estimation process such as the choice of input parameter
values may also affect the resultant D (discussed below).
Several researchers (Gallant et al. 1994, Tate 1998, Lam et al. 2002, Sun 2006)
have attempted to gauge the direction and magnitude of errors introduced by
existing computational methods. Their results show that no single method appears
able to produce accurate estimates of D over the entire range of D. For example,
Lam et al. (2002) found that the triangular prism method was most accurate for
images having higher spatial complexity, for images where generated D52.5 the
isarithm returned best results, and the variogram method was a comparatively poor
estimator for all surfaces. These results suggest that a particular fractal
computational method may be better suited to certain images than others,
depending largely on the roughness of the image to be analysed. Practical use of
such research findings may be challenged by the fact that relatively little is known
about the actual ranges of D of different types of remotely sensed images.
Furthermore, determination of the degree of ‘roughness’ of an image would
represent a priori knowledge about the textural complexity of the image. Can other
existing texture analysis techniques, such as co-occurrence matrices (Haralick et al.
1973), wavelets (Mallat 1989), and local variance (Woodcock and Strahler 1987), be
used for this purpose? What kind of relationship exists between other measures of
texture and the D? The existing literature does not seem to provide satisfactory
answers to these questions. As such, they constitute a fertile field for future research.
5.3 Parameter specifications and the computed D
Implementation of fractal computational methods requires the user to define a
certain number of input parameters, such as beginning and ending step sizes,
interval spacing (step size), direction of computation, and so forth. The choice of
such parameter values is an important issue as these decisions can greatly influence
the computed D. In the methods discussed above (table 1), the estimate of D is
derived as some function of the slope of the regression lines and the slope is
determined using the least-squares method. This means that the selection of the
smallest and largest step sizes and the interval spacing is a critical decision as these
parameter values may affect the slope of the regression line and hence the computed
4980 W. Sun et al.
D. However, there are no established guidelines for choosing the beginning orending step size. Shelberg et al. (1982) proposed to use one-half of the average
distance between every pair of adjacent points as the smallest step size. Andrle
(1992), on the other hand, suggested that the smallest step size be twice the shortest
distance between any two points. In practice, the smallest step size is often chosen to
be close to the limiting resolution of the datasets used (Clarke and Schweizer 1991).
As for the largest step size, several authors have suggested using a reduced largest
step size for two reasons. First, use of a smaller largest step size may help minimize
the effects of partial steps (Andrle 1992). Second, when working with self-affineprofiles, it is critical to ensure that the largest step size remains smaller than the
crossover length (Mandelbrot 1985).
It is widely accepted that for statistical reasons, step size should increase as a
power of two (i.e. doubling the step size) so that data points on the log–log plot are
equally spaced in their independent variable. Doubling the step size will, however,
rapidly cover the data and may result in too few data points available for regression
analysis if the image being analysed is not sufficiently large. The number of step sizes
(or step size increments) is another parameter that needs to be selected with care asthis parameter determines the number of data points available for the regression
analysis. To obtain reliable results, it is necessary to use a sufficiently large number
of step sizes. However, there is no established guideline for choosing an
‘appropriate’ number of step sizes either. Shelberg et al. (1982) suggested that five
to eight step size increments be used. It should be noted that the need to use a large
number of data points in regression analysis is in conflict with the sampling strategy
of doubling the step size, since doubling the step size means that fewer step size
increments can be made (Klinkenberg 1994). As such, careful consideration isneeded in selecting the beginning and ending step size and the interval spacing.
5.4 Input data and the computed D
Several factors related to the remote sensing data used in an analysis may also affect
the computed D. We outline below some of the factors that require attention in
computing the D of remotely sensed images.
N To the extent that the image being measured is not strictly self-similar, the
spatial resolutions, regions, directions, and sampling process selected as input
data may all affect the computed D. It may be desirable that in reporting
research findings, the researcher gives sufficient details about the input data
used. Without such information, comparison of results from different studies
would be meaningless.
N Textures of the same land cover feature in different spectral bands are often
different in terms of contrast, smoothness, spatial variation, and so forth.Therefore, it is most likely that computed D values will vary greatly with the
bands chosen. Selecting bands with the ‘right’ spectral information content will
be an important task in fractal analysis of remotely sensed images, if the
technique is to be effective in capturing the textural characteristics of the
phenomena under consideration.
N The way in which an image is represented, such as the range of pixel values
used, may also have an effect on the computed D when using certain
computational methods. Lam et al. (2002), for example, found that thetriangular prism method was particularly sensitive to changes in image contrast
and recommended that images be normalized before computing D.
Fractal analysis of remotely sensed images 4981
N The quality of the image used may also affect the estimated D as the
‘contribution’ of noise to the spatial variations in pixel values could be
significant. An assessment of image quality using such techniques as the signal-
to-noise ratios may be useful (De Jong and Burrough 1995, Emerson et al.
1999).
5.5 Computing local D values
A major contribution of De Jong and Burrough’s (1995) work reviewed above is
that they demonstrated how to locally compute the D of real remotely sensed
images. This is an important step as most existing methods only yield a lumped D
value for the entire image. While such global D values may be useful for certain
purposes, they cannot be used to segment images. Surprisingly few studies have
attempted to systematically calculate local D values and use such information to
classify real images. How to develop efficient algorithms to compute local D values
from real images and use local D values as textural information to improve image
classification is an area that holds great potential for future research.
As De Jong and Burrough (1995) have shown, local D values can be computed
using moving window techniques. As with all other techniques involving kernels,
computation of local D values must deal with two undesirable effects, i.e. the
blurring effect and the boundary effect. In their study, De Jong and Burrough (1995)
found that the blurring effects resulted from a 9 by 9 analysis window were visible.
Using local windows to compute D also means that local D values will not be
available for certain rows and columns, whose number equals to one half of the local
window size used, around the edge of the image being analysed.
A potentially more challenging issue in computing local D values is the choice of
window size. On one hand, minimization of local window size is required to capture
details of local variations in land covers. On the other hand, the regression technique
used in the estimation of D requires a sufficiently large number of step sizes to be
available within an analysis window. This, coupled with the sampling strategy of
doubling the step size, means that the window size will be quite large. For example,
if the number of step sizes is chosen to be five, this will translate into a local window
size of 17 by 17 pixels. Use of large window size may, however, lead to several
problems. First, a large window will include more land covers and it can lead to
mixed pixel problems. Second, using a large window means that land cover features
smaller than the window size will not be identified in classification. Third, using a
large window will also lead to loss of more pixels on the edges (i.e. boundary effects).
As such, how to choose an ‘appropriate’ window size for computing local D values is
an issue that deserves further research.
5.6 How to better describe image texture using fractal geometry?
A major finding from the research published to date is that, while fractal dimension
appears able to capture certain aspects of the surface properties of remotely sensed
images, use of D alone cannot sufficiently describe image textures and achieve
satisfactory classification results. Several approaches have been suggested to address
this problem. Below we briefly discuss three such approaches.
5.6.1 Multi-parameter fractal description of image texture. Most existing studies
applying fractal techniques to image texture analysis have concentrated on a single
4982 W. Sun et al.
fractal parameter, i.e. the fractal dimension. Certain computational methods, such
as the variogram and the Fourier power spectral methods, produce more than just
one parameter. Variogram analysis, for example, generates not only the estimate of
the slope but estimates for the range and the intercept of a variogram (Chen and
Gong 2004). Studies in other fields such as geomorphology have shown that the log–
log plot ordinate intercept of a variogram seems to capture certain information that
is not captured by D (Klinkenberg 1992). Therefore, it seems desirable to use more
fractal parameters to characterize image textures instead of using only D. A
disadvantage of this multi-parameter fractal approach is that not all methods can
provide parameters other than D.
5.6.2 Multifractal models. All the methods reviewed in this paper are based on a
mono-fractal approach, which assumes that the object under consideration can be
characterized by a single fractal dimension. Evidence from the geosciences suggests,
however, that the various natural processes (geological, geomorphological,
ecological, etc) operating at different scales do not have the same influence on the
structures in nature (Mark and Aronson 1984, Roy et al. 1987, Klinkenberg and
Goodchild 1992). As a result, most natural phenomena are characterized by
different dominant structures at different scales (Goodchild and Mark 1987, Feder
1988, Mandelbrot 1989, Milne 1991, Meakin 1991). Analysis of real remotely sensed
data has also shown that the scaling behaviour of image properties deviates greatly
from the ideal mono-fractal dimension assumption (De Cola 1993). This suggests
that multifractal models appear to be more suited to characterization of image
textures. There is a growing body of literature on the application of multifractal
models in image analysis (e.g. Peitgen and Saupe 1988, Arduini et al. 1992, De Cola
1993, Fioravanti 1994, Cheng 1999, Parrinello and Vaughan 2002, Posadas et al.
2005). A detailed discussion of this literature is beyond the scope of this paper.
5.6.3 Lacunarity and image texture analysis. Fractal dimension is a parameter that
measures the geometric complexity of the shape of an object. Obviously, shape is not
the only property of image texture. Other factors such as the size and distribution of
a textural feature and its spatial relations to other features may also play an
important part in differentiating one type of texture from another. This means that
use of D alone may not be sufficient in characterization of image textures.
Mandelbrot (1982) pointed out that different fractal sets may share the same D and
yet have strikingly different appearances or textures. This seems to have been borne
out in several empirical studies in which the estimated D values of different land
cover types were found to be overlapping or even the same (Keller et al. 1989, De
Jong and Burrough 1995, Myint 2003). Lacunarity analysis is a technique
introduced by Mandelbrot (1982) to deal with fractal objects of the same dimension
with different textural appearances. Lacunarity is related to the distribution of gap
sizes: low lacunarity objects are characterized by similar or same gap sizes and
therefore appear homogeneous, whereas high lacunarity objects are heterogeneous
(Mandelbrot 1982, Allain and Cloitre 1991, Dong 2000a). Although originally
developed for fractal objects, lacunarity analysis has been proposed to be a general
technique for the analysis of nonfractal and multifractal patterns (Plotnick et al.
1996). Image segmentation using lacunarity or a combination of D and lacunarity
seem to have obtained good results (e.g., Keller et al. 1989, Dong 2000b). It appears
that incorporating lacunarity measures into the classification procedure is another
promising approach to enhancing classification results. Readers interested in the
Fractal analysis of remotely sensed images 4983
methods for computing lacunarity are directed to Mandelbrot (1982), Allain andCloitre (1991) and Dong (2000b).
6. Conclusions
Fractal geometry appears to provide a useful tool for characterizing textural
features in remotely sensed images because most of what we measure in remotesensing—boundaries of land covers, patches of landscapes, rivers and water bodies,
tree crowns, etc—is discontinuous, complex, and fragmented. Fractal techniques
have been applied to measure the ‘roughness’ or geometric complexity of land
surface features in unclassified and classified images. Quantitative information
about local variations in estimated D values has been used as a texture measure to
segment remotely sensed images. Fractal techniques have also been used to
investigate the scaling behaviour of environmental phenomena and the results from
this stream of research may prove valuable for choosing ‘optimal’ resolutions forstudying environmental phenomena at different scales in remote sensing and GIS.
Despite the potential utility of fractal techniques, several methodological and
practical measurement problems have been encountered in fractal analysis of
remotely sensed images. For example, the computed D is supposed to capture the
differences in the characteristics of image textures. However, a host of factors other
than image texture, such as the computational method used, the choice of input
parameter values, input images, and so forth, may all have an effect on the
computed D. As a result, it seems extremely difficult, if possible at all, to determinewhether the observed differences in computed D values is a result of true differences
in image texture or a result of certain arbitrary decisions made during the estimation
process. As such, the question what actually is measured in the computed D remains
unanswered.
For D to be a useful parameter, the methods of computing D must be robust,
consistent, and have the ability to distinguish visually different textures. Research
published to date has shown, however, that significant variations in computed D can
be introduced by computational methods. Therefore, the choice of method is animportant issue in the computation of D for remotely sensed data. Researchers need
to be aware of the comparative performance of different methods proposed in the
literature and the biases that are associated with a particular method. Blind use of a
method without knowing its applicable D ranges and potential errors may lead to
poor or even erroneous results. Given the uncertainty surrounding the nature and
extent of method-induced errors, more systematic evaluation of existing computa-
tional methods is needed.
A major drawback in using fractal techniques for analysing remotely sensed
images is that they can be applied only to single bands{. Since real remotely sensedimages are generally multispectral ones, it appears desirable to develop what may be
called ‘multivariate fractal methods.’ Such multivariate fractal methods should
enable the analysis of all bands together and, therefore, would represent a
tremendous improvement to the existing methods in fractal analysis of remotely
sensed data.
For the most part, existing research applying fractal techniques to remote sensing
problems rests on the assumption that image textures can be described by a single
(mono-fractal) dimension. Evidence from remote sensing applications as well as
{ We thank an anonymous reviewer for bringing this issue to our attention.
4984 W. Sun et al.
other fields of the geosciences suggests that the structures underlying most naturalphenomena are most likely multifractals. This implies that fractal analysis of
remotely sensed images without checking their dimensionalities may be problematic.
More important, further research is needed to determine whether multifractal
models could do a better job in characterizing image textures.
The fractal dimension can be thought of as a summary statistic measuring the
overall geometric complexity of image textures. Like many summary statistics, the
fractal dimension is obtained by ‘averaging’ local tonal variations and it only
captures one aspect of the spatial variations of grey levels in the image. Theestimated D of a textural feature, for example, tells us nothing about its actual size
or its spatial distribution, nor can we infer its spatial relations to other textural
features from its D alone. This may explain why many studies have found that,
despite its usefulness, use of D alone is insufficient to accurately describe image
textures and achieve satisfactory classification results. It appears that the utility of D
may be explored to a fuller extent when it is used in conjunction with other texture
measures and perhaps spectral classification approaches as well.
Acknowledgements
We would like to thank three anonymous reviewers for their very helpful comments
on an earlier draft of this paper.
ReferencesALLAIN, C. and CLOITRE, M., 1991, Characterizing the lacunarity of random and determined
fractal set. Physics Review A, 44, pp. 3552–3558.
ANDRLE, R., 1992, Estimating fractal dimension with the divider method in geomorphology.
Geomorphology, 5, pp. 131–141.
ARDUINI, F., FIORAVANTI, S. and GIUSTO, D.D., 1992, On computing multifractality for
texture discriminiation. In Proceedings of EUSIPCO-92, 24–27 August 1992, Brussels,
Belgium, J. Vandewelle and R. Boite (Eds) (Brussels: Elsevier Science), pp. 1457–1461.
ARMSTRONG, A., 1986, On the fractal dimensions of some transient soil properties. Journal of
Soil Science, 37, pp. 641–652.
BARNSLEY, M.F., 1989, Iterated function systems. In Chaos and Fractals: Mathematics behind
the Computer, R.L. Devaney and L. Keen (Eds), pp. 127–144 (Providence, RI:
American Mathematical Society).
BATTY, M. and LONGLEY, P.A., 1986, The fractal simulation of urban structure. Environment
and Planning A, 18, pp. 1143–1179.
BELLOULATA, K. and KONRAD, J., 2002, Fractal image compression with region-based
functionality. IEEE Transactions on Image Processing, 11, pp. 351–362.
BLACHER, S., BROUERS, F. and VAN DYCK, R., 1993, On the use of fractal concepts in image
analysis. Physica A, 197, pp. 516–527.
BROWN, S.R., 1995, Measuring the dimension of self-affine fractals: examples of rough
surfaces. In Fractals in the Earth Sciences, C.C. Barton and P.R. La Pointe (Eds), pp.
77–87 (New York: Plenum Press).
BURROUGH, P.A., 1981, Fractal dimensions of landscapes and other environmental data.
Nature, 294, pp. 240–242.
BURROUGH, P.A., 1993, Fractals and geostatistical methods in landscape studies. In Fractals
in Geography, N.S.-N. Lam and L. De Cola (Eds), pp. 87–121 (New Jersey: Prentice
Hall).
CALVET, L. and FISHER, A., 2002, Multifractality in asset returns: Theory and practice.
Reviews of Economics and Statistics, 84, pp. 381–406.
CARR, J. and BENZER, W., 1991, On the practice of estimating fractal dimension.
Mathematical Geology, 23, pp. 945–958.
Fractal analysis of remotely sensed images 4985
CHAUDHURI, B.B., SARKAR, N. and KUNDU, P., 1993, Improved fractal geometry based
texture segmentation technique. IEEE Proceedings E, 140, pp. 233–241.
CHEN, Q. and GONG, P., 2004, Automatic variogram parameter extraction for textural
classification of IKONOS imagery. IEEE Transactions on Geoscience and Remote
Sensing, 42, pp. 1106–1115.
CHEN, C.C., DAPONTE, J.S. and FOX, M.D., 1989, Fractal feature analysis and classification
in medical imaging. IEEE Transactions on Medical Imaging, 6, pp. 133–142.
CHENG, Q., 1999, Multifractality and spatial statistics. Computers and Geosciences, 25, pp.
949–961.
CLARKE, K.C., 1986, Computation of the fractal dimension of topographic surfaces using the
triangular prism surface area method. Computers and Geosciences, 12, pp. 713–722.
CLARKE, K.C. and SCHWEIZER, D.M., 1991, Measuring the fractal dimension of natural
surfaces using a robust fractal estimator. Cartography and Geographic Information
Systems, 18, pp. 37–47.
CLIFF, A.D. and ORD, J.K., 1973, Spatial Autocorrelation (London: Pion Limited).
DE COLA, L., 1989, Fractal Analysis of a classified Landsat scene. Photogrammetric
Engineering and Remote Sensing, 55, pp. 601–610.
DE COLA, L., 1993, Multifractals in image processing and process imaging. In Fractals in
Geography, N.S.-N. Lam and L. De Cola (Eds), pp. 282–304 (New Jersey: Prentice
Hall).
DE JONG, S.M., 1993, An application of spatial filtering techniques for land cover mapping
using TM images. Geocarto International, 8, pp. 43–49.
DE JONG, S.M. and BURROUGH, P.A., 1995, A fractal approach to the classification of
Mediterranean vegetation types in remotely sensed images. Photogrammetric
Engineering and Remote Sensing, 61, pp. 1041–1053.
DENNIS, T.J. and DESSIPRIS, N.G., 1989, Fractal modeling in image texture analysis. IEEE
Proceedings, 136, pp. 227–235.
DEVANEY, R.L. and KEEN, L., 1989, Chaos and Fractals: Mathematics Behind the Computer
(Providence, RI: American Mathematical Society).
DONG, P.L., 2000a, Lacunarity for spatial heterogeneity measurement in GIS. Geographic
Information Sciences, 6, pp. 20–26.
DONG, P.L., 2000b, Test of a new lacunarity estimation method for image texture analysis.
International Journal of Remote Sensing, 17, pp. 3369–3373.
DYSON, F., 1978, Characterizing irregularity. Science, 200, pp. 677–678.
EMERSON, C.W., LAM, N.S.-N. and QUATTROCHI, D.A., 1999, Multiscale fractal analysis of
image texture and pattern. Photogrammetric Engineering and Remote Sensing, 65, pp.
51–61.
EMERSON, C.W., QUATTROCHI, D.A. and LAM, N.S.-N., 2004, Spatial metadata for remote
sensing imagery. Available online at: www.esto.nasa.gov/conferences/estc2004/papers/
b8p2.pdf (accessed 10 May 2005).
FEDER, J., 1988, Fractals (New York: Plenum Press).
FIORAVANTI, S., 1994, Multifractals: theory and application to image texture recognition. In
Fractals in Geoscience and Remote Sensing, Proceedings of A Joint JRC/EARSeL
Expert Meeting, pp. 152–175 (Luxemburg: European Commission).
FOURNIER, A., FUSSELL, D. and CARPENTER, L., 1982, Computer rendering of stochastic
models. Communications of the ACM, 25, pp. 371–384.
FOX, C. and HAYES, D.E., 1985, Quantitative methods for analyzing the roughness of the
seafloor. Reviews of Geophysics, 23, pp. 1–48.
GALLANT, J.C., MOORE, I.D., HUTCHINSON, M.F. and GESSLER, P., 1994, Estimating fractal
dimension of profiles: A comparison of methods. Mathematical Geology, 26, pp.
455–481.
GAO, J. and XIA, Z.G., 1996, Fractals in physical geography. Progress in Physical Geography,
20, pp. 178–191.
4986 W. Sun et al.
GERMAIN, M., BENIE, G.B., BOUCHER, J.M., FOUCHER, S., FUNG, K. and GOITA, K., 2003,
Contribution of the fractal dimension to multiscale adaptive filtering of SAR imagery.
IEEE Transactions on Geosciences and Remote Sensing, 41, pp. 1765–1772.
GHAZEL, M., FREEMAN, G.H. and VRSCAY, E.R., 2003, Fractal image denoising. IEEE
Transactions on Image Processing, 12, pp. 1560–1578.
GONG, P. and HOWARTH, P.J., 1990, The use of structural information for improving land-
cover classification accuracies at the rural-urban fringe. Photogrammetric Engineering
and Remote Sensing, 56, pp. 67–73.
GONG, P., MARCEAU, D.J. and HOWARTH, P.J., 1992, A comparison of spatial feature
extraction algorithms for land-use classification with SPOT HRV data. Remote
Sensing of Environment, 40, pp. 137–151.
GOODCHILD, M.F., 1980, Fractals and the accuracy of geographical measures. Mathematical
Geology, 12, pp. 85–98.
GOODCHILD, M.F. and MARK, D.M., 1987, The fractal nature of geographic phenomena.
Annals of the Association of American Geographers, 77, pp. 265–278.
GREEN, T.R. and ERSKINE, R.H., 2004, Measurement, scaling, and topographic analyses
of spatial crop yield and soil water content. Hydrological Processes, 18, pp.
1447–1465.
HARALICK, R.M., SHANMUGAN, K. and DINSTEIN, I., 1973, Texture features for image
classification. IEEE Transactions on Systems, Man, and Cybernetics, SMC–3, pp.
610–621.
HE, H.S., VENTURA, S.J. and MLADENOFF, D.J., 2002, Effects of spatial aggregation
approaches on classified satellite imagery. International Journal of Geographical
Information Science, 16, pp. 93–109.
JAGGI, S., QUATTROCHI, D.A. and LAM, N.S.-N., 1993, Implementation and operation of
three fractal measurement algorithms for analysis of remote-sensing data. Computer
and Geosciences, 19, pp. 745–767.
JONES, J., THOMAS, R. and EARWICKER, P., 1989, Fractal properties of computer-generated
and natural geophysical data. Computers and Geosciences, 15, pp. 227–235.
KELLER, J.M., CROWNOVER, R.M. and CHEN, R.Y., 1987, Characteristics of natural scenes
related to the fractal dimension. IEEE Transactions on Pattern Analysis and Machine
Intelligence, 9, pp. 621–627.
KELLER, J.M., CHEN, S. and CROWNOVER, R.M., 1989, Texture description and segmentation
through fractal geometry. Computer Vision, Graphics, and Image Processing, 45, pp.
150–166.
KENNEDY, S.K. and LIN, W., 1986, FRACT—A Fortran subroutine to calculate the variables
necessary to determine the fractal dimension of closed forms. Computers and
Geosciences, 12, pp. 705–712.
KLINKENBERG, B., 1992, Fractal and morphometric measures: Is there a relationship?
Geomorphology, 5, pp. 5–20.
KLINKENBERG, B., 1994, A review of methods used to determine the fractal dimension of
linear features. Mathematical Geology, 26, pp. 23–46.
KLINKENBERG, B. and GOODCHILD, M.F., 1992, The fractal properties of topography: A
comparison of methods. Earth Surface Processes and Landforms, 17, pp. 217–234.
LAGRO JR, J., 1991, Assessing patch shape in landscape mosaics. Photogrammetric
Engineering and Remote Sensing, 57, pp. 285–293.
LAM, N.S.-N., 1990, Description and measurement of Landsat TM images using fractals.
Photogrammetric Engineering and Remote Sensing, 56, pp. 187–195.
LAM, N.S.-N. and DE COLA, L., 1993a, Fractals in Geography New Jersey: Prentice Hall).
LAM, N.S.-N. and DE COLA, L., 1993b, Fractal measurement. In Fractals in Geography,
N.S.-N. Lam and L. De Cola (Eds), pp. 23–55 (New Jersey: Prentice Hall).
LAM, N.S.-N., QIU, H.L., QUATTROCHI, D.A. and EMERSON, C.W., 2002, An evaluation of
fractal methods for characterizing image complexity. Cartography and Geographic
Information Science, 29, pp. 25–35.
Fractal analysis of remotely sensed images 4987
LEE, W.L., CHEN, Y.C. and HSIEH, K.S., 2003, Ultrasonic liver tissues classification by fractal
feature vector based on M-band wavelet transform. IEEE Transactions on Medical
Imaging, 22, pp. 382–392.
LIANG, S., 2004, Quantitative Remote Sensing of Land Surfaces (New York: John Wiley &
Sons).
LOEHLE, C., 1983, The fractal dimension and ecology. Speculations in Science and Technology,
6, pp. 131–142.
LORIMER, N.D., HAIGHT, R.G. and LEARY, R.A., 1994, The Fractal Forest: Fractal Geometry
and Applications in Forest Science (USDA: General Technical Report NC–170).
LOVEJOY, S., 1982, Area-perimeter relation for rain and cloud areas. Science, 216, pp.
185–187.
LOVEJOY, S. and SCHERTZER, D., 1985, Generalized scale invariance in the atmosphere and
fractal models of rain. Water Resources Research, 21, pp. 1233–1250.
LOVEJOY, S. and SCHERTZER, D., 1990, Fractals, rain drops, and resolution dependence of
rain measurements. Journal of Applied Meteorology, 29, pp. 1167–1170.
LU, S.Z. and HELLAWELL, A., 1995, Using fractal analysis to describe irregular
microstructures. Journal of Materials, 47, pp. 14–16.
MAITRE, H. and PINCIROLI, M., 1999, Fractal characterization of a hydrological basin using
SAR satellite images. IEEE Transactions on Geosciences and Remote Sensing, 37, pp.
175–181.
MALLAT, S.G., 1989, A theory for multi-resolution signal decomposition: The wavelet
representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11,
pp. 674–693.
MANDELBROT, B.B., 1967, How long is the coast of Britain? Statistical self-similarity and
fractional dimension. Science, 156, pp. 636–638.
MANDELBROT, B.B., 1977, Fractals: Form, Chance and Dimension (San Francisco, CA: W.H.
Freeman and Company).
MANDELBROT, B.B., 1982, The Fractal Geometry of Nature (New York: W.H. Freeman and
Company).
MANDELBROT, B.B., 1985, Self-affine fractals and the fractal dimension. Physica Scripta, 32,
pp. 257–260.
MANDELBROT, B.B., 1989, Multifractal measures, especially for the geophysicists. Pageoph,
131, pp. 5–42.
MANDELBROT, B.B. and HUDSON, R.L., 2004, The (mis)Behavior of Markets: A fractal view
of risk, ruin, and reward (New York: Basic Books).
MARK, D.M. and ARONSON, P.B., 1984, Scale-dependent fractal dimensions of topographic
surfaces: An empirical investigation with application in geomorphology and computer
Mapping. Mathematical Geology, 16, pp. 671–683.
MEAKIN, P., 1991, Fractal aggregates in geophysics. Reviews of Geophysics, 29, pp. 317–354.
MILNE, B.T., 1991, Lessons from applying fractal models to landscape patterns. In
Quantitative Methods in Landscape Ecology, M.G. Turner and R.H. Gardern (Eds),
pp. 199–235 (New York: Springer Verlag).
MYINT, S.W., 2003, Fractal approaches in texture analysis and classification of remotely
sensed data: Comparisons with spatial autocorrelation techniques and simple
descriptive statistics. International Journal of Remote Sensing, 24, pp. 1925–1947.
NORMANT, F. and TRICOT, C., 1993, Fractal simplification of lines using convex hulls.
Geographical Analysis, 25, pp. 118–129.
ORFORD, J.D. and WHALLEY, W.B., 1983, The use of the fractal dimension to quantify the
morphology of irregular-shaped particles. Sedimentology, 30, pp. 655–668.
PACHEPSKY, Y.A., RITCHIE, J.C. and GIMENEZ, D., 1997, Fractal modeling of airborne laser
altimetry data. Remote Sensing of Environment, 61, pp. 150–161.
PARRINELLO, T. and VAUGHAN, R.A., 2002, Multifractal analysis and feature extraction in
satellite imagery. International Journal of Remote Sensing, 23, pp. 1799–1825.
4988 W. Sun et al.
PEITGEN, H.O. and RICHTER, P.H., 1986, The Beauty of Fractals (New York: Springer
Verlag).
PEITGEN, H.O. and SAUPE, D., 1988, The Science of Fractal Images (New York: Springer
Verlag).
PELEG, S., NAOR, J., HARTLEY, R. and AVNIR, D., 1984, Multiple resolution texture analysis
and classification. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6,
pp. 518–528.
PENTLAND, A.P., 1984, Fractal-based descriptions of natural scenes. IEEE Transactions on
Pattern Analysis and Machine Intelligence, 6, pp. 661–674.
PIECH, M.A. and PIECH, K.R., 1990, Fingerprints and fractal terrain. Mathematical Geology,
22, pp. 457–485.
PLOTNICK, R.E., GARDNER, R.H., HARGROVE, W.W., PRESTEGARRD, K. and
PERLMUTTER, M., 1996, Lacunarity analysis: A general technique for the analysis
of spatial patterns. Physical Review E, 53, pp. 5461–5468.
POSADAS, A.N.D., QUIROZ, R., ZOROGASTUA, P.E. and LEON-VELARDE, C., 2005,
Multifractal characterization of the spatial distribution of ulexite in a Bolivian salt
flat. International Journal of Remote Sensing, 26, pp. 615–627.
PRATT, W.K., FAUGERAS, O.D. and GAGALOWICZ, A., 1978, Visual discrimination of
stochastic texture fields. IEEE Transactions on Systems, Man and Cybernetics, SMC–
8, pp. 796–804.
QIU, H., LAM, N.S.-N., QUATTOCHI, D.A. and GAMON, J.A., 1999, Fractal characterization of
hyperspectral imagery. Photogrammetric Engineering and Remote Sensing, 65, pp.
63–71.
READ, J.M. and LAM, N.S.-N., 2002, Spatial methods for characterizing land cover and
detecting land-cover changes for the tropics. International Journal of Remote Sensing,
23, pp. 2457–2474.
RICHARDSON, L.F., 1961, The problem of contiguity: An appendix to ‘‘Statistics of Deadly
Quarrels.’’ General Systems Yearbook, 6, pp. 139–187.
RICOTTA, C. and AVENA, G.C., 1998, Fractal modeling of the remotely sensed two-
dimensional net primary production pattern with annual cumulative AVHRR NDVI
data. International Journal of Remote Sensing, 19, pp. 2413–2418.
RICOTTA, C., AVENA, G.C., OLSEN, E.R., RAMSEY, R.D. and WINN, D.S., 1998, Monitoring
the landscape stability of Mediterranean vegetation in relation to fire with a fractal
algorithm. International Journal of Remote Sensing, 19, pp. 871–881.
ROY, A.G., GRAVEL, G. and GAUTHIER, C., 1987, Measuring the dimension of surfaces: A
review and appraisal of different methods. Proceedings of the Eighth International
Symposium on Computer-Assisted Cartography (Auto–Carto 8), pp. 68–77.
SARKAR, N. and CHAUDHURI, B.B., 1992, An efficient approach to estimate fractal dimension
of textural images. Pattern Recognition, 25, pp. 1035–1041.
SHELBERG, M.C., MOELLERING, H. and LAM, N.S.-N., 1982, Measuring the fractal dimension
of empirical cartographic curves. Proceedings of the Fifth International Symposium on
Computer-Assisted Cartography (Auto-Carto 5), pp. 481–490.
SHELBERG, M.C., LAM, N.S.-N. and MOELLERING, H., 1983, Measuring the fractal dimension
of surfaces. Proceedings of the Sixth International Symposium on Computer-Assisted
Cartography (Auto–Carto 6), pp. 319–328.
SUN, W., 2005, Three new implementations of the triangular prism method for computing the
fractal dimension of remote sensing images. Photogrammetric Engineering and Remote
Sensing, 72, pp. 373–382.
SUN, W., KOLAPPAL, A.Z. and GONG, P., 2006, Two computational methods for detecting
anisotropy in image texture. Geographic Information Science (forthcoming).
TATE, N.J., 1998, Estimating the fractal dimension of synthetic topographic surfaces.
Computers and Geosciences, 24, pp. 325–334.
TSO, B. and MATHER, P.M., 2001, Classification Methods for Remotely Sensed Data (London:
Taylor and Francis).
Fractal analysis of remotely sensed images 4989
TURCOTTE, D.L., 1992, Fractals and Chaos in Geology and Geophysics (Cambridge:
Cambridge University Press).
VOSS, R., 1988, Fractals in nature: From characterization to simulation. In The Science of
Fractal Images, H.O. Peitgen and D. Saupe (Eds), pp. 21–70 (New York: Springer
Verlag).
WANG, L. and HE, D.C., 1990, A new statistical approach for texture analysis.
Photogrammetric Engineering and Remote Sensing, 56, pp. 61–66.
WENG, Q., 2003, Fractal analysis of satellite-detected urban heat island effect.
Photogrammetric Engineering and Remote Sensing, 69, pp. 555–566.
WESZKA, J., DYER, C.R. and ROSENFELD, A., 1976, A comparative study of texture measures
for terrain classification. IEEE Transactions on Systems, Man, and Cybernetics, SMC-
6, pp. 269–285.
WIENS, J.A., 1989, Spatial scaling in ecology. Functional Ecology, 3, pp. 385–397.
WOODCOCK, C.E. and STRAHLER, A.H., 1987, The factor of scale in remote sensing. Remote
Sensing of Environment, 21, pp. 311–332.
WU, C.M., CHEN, Y.C. and HSIEH, K.S., 1992, Texture features for classification of ultrasonic
liver images. IEEE Transactions on Medical Imaging, 11, pp. 141–152.
4990 Fractal analysis of remotely sensed images
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